Problem: Describe the solutions to the following quadratic equation: $4x^{2}+5x+1 = 0$
We could use the quadratic formula to solve for the solutions and see what they are, but there's a shortcut... $ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ Think about what the part of the quadratic formula under the radical tells us about the solutions. Substitute the $a$ $b$ , and $c$ coefficients from the quadratic equation: $ \begin{array} && b^2-4ac \\ \\ =& 5^2 - 4 ( 4)(1) \\ \\ =& 9 \end{array} $ Because ${b^2 - 4ac}$ is a perfect square, its square root is rational and the quadratic formula reduces to $\dfrac{-b \pm 3}{2a} $, which means there are two rational solutions.